A-M-M

231. A-M-M

Mult. trend, mult. seasonality

ETS (𝐴, 𝑀, 𝑀)

𝑥_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} 𝑠_{𝑡 - 𝑚} + 𝜀_{𝑡}

𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / (𝑙_{𝑡 - 1} 𝑠_{𝑡 - 𝑚})

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡} / (𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1})

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} 𝑏_{𝑡}^{ℎ} 𝑠_{𝑡 + ℎ - 𝑚_{ℎ}^{+}}

Example:

ETS (𝐴, 𝑀, 𝑀)

Given

Smoothing parameters: $𝛼 = 0.5$ , $𝛽 = 0.4$ , $𝛾 = 0.2$
Initial states: $𝑙_{0} = 12$ , $𝑏_{0} = 1$ , $(𝑠_{- 3}, 𝑠_{- 2}, 𝑠_{- 1}, 𝑠_{0}) = (1.2, 1, 0.8, 1)$ , seasonal period $𝑚 = 4$
Data:

$𝑡$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
$x_{𝑡}$	12	10	8	11	14	12	9	13	16	14	11	15	18	16	13	17

Step 1 — formula

Observation:

𝑥_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} 𝑠_{𝑡 - 𝑚} + 𝜀_{𝑡}

Conditional mean (one-step-ahead forecast ${\hat{𝑥}}_{𝑡 | 𝑡 - 1} = 𝜇_{𝑡}$ ):

𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} 𝑠_{𝑡 - 𝑚}

Innovation:

𝜀_{𝑡} = 𝑥_{𝑡} - 𝜇_{𝑡}

State updates:

𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}

𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / (𝑙_{𝑡 - 1} 𝑠_{𝑡 - 𝑚})

𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡} / (𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1})

Forecast $ℎ$ steps ahead from time $𝑡$ (using current-period states):

{\hat{𝑥}}_{𝑡 + ℎ | 𝑡} = 𝑙_{𝑡} 𝑏_{𝑡}^{ℎ} 𝑠_{𝑡 + ℎ - 𝑚_{ℎ}^{+}}

where $ℎ \in {1, 2, 3, \dots}$ is the forecast horizon (how many steps ahead); $𝑚_{ℎ}^{+} = ((ℎ - 1) mod 𝑚) + 1$ picks the right seasonal slot for the period $ℎ$ steps ahead (cycles through $1, 2, \dots, 𝑚$ ).

Step 2 — apply at $𝑡 = 1$

𝜇_{1} = 12 \cdot 1 \cdot 1.2 = 14.4

𝜀_{1} = 𝑥_{1} - 𝜇_{1} = 12 - 14.4 = - 2.4

𝑙_{1} = 12 \cdot 1 + 0.5 \cdot (- 2.4) / 1.2 = 11

𝑏_{1} = 1 + 0.4 \cdot (- 2.4) / (12 \cdot 1.2) = 0.9333

𝑠_{1} = 1.2 + 0.2 \cdot (- 2.4) / (12 \cdot 1) = 1.16

Step 3 — iterate

Each column header is the equation that produced its values. Values rounded to 4 decimal places; arithmetic performed at full precision.

$𝑡$	$𝑥_{𝑡}$	$𝜇_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} 𝑠_{𝑡 - 𝑚}$	$𝜀_{𝑡}$	$𝑙_{𝑡} = 𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1} + 𝛼 𝜀_{𝑡} / 𝑠_{𝑡 - 𝑚}$	$𝑏_{𝑡} = 𝑏_{𝑡 - 1} + 𝛽 𝜀_{𝑡} / (𝑙_{𝑡 - 1} 𝑠_{𝑡 - 𝑚})$	$𝑠_{𝑡} = 𝑠_{𝑡 - 𝑚} + 𝛾 𝜀_{𝑡} / (𝑙_{𝑡 - 1} 𝑏_{𝑡 - 1})$
1	12	$14.4$	$- 2.4$	$11$	$0.9333$	$1.16$
2	10	$10.2667$	$- 0.2667$	$10.1333$	$0.9236$	$0.9948$
3	8	$7.4876$	$0.5124$	$9.6798$	$0.9489$	$0.8109$
4	11	$9.1853$	$1.8147$	$10.0927$	$1.0239$	$1.0395$
5	14	$11.9874$	$2.0126$	$11.2015$	$1.0927$	$1.199$
6	12	$12.1759$	$- 0.1759$	$12.1511$	$1.0864$	$0.9919$
7	9	$10.7049$	$- 1.7049$	$12.1493$	$1.0172$	$0.7851$
8	13	$12.8459$	$0.1541$	$12.4317$	$1.022$	$1.042$
9	16	$15.2334$	$0.7666$	$13.0253$	$1.0426$	$1.211$
10	14	$13.4706$	$0.5294$	$13.8471$	$1.059$	$0.9997$
11	11	$11.5129$	$- 0.5129$	$14.3373$	$1.0401$	$0.7781$
12	15	$15.5389$	$- 0.5389$	$14.6539$	$1.0257$	$1.0348$
13	18	$18.202$	$- 0.202$	$14.9469$	$1.0211$	$1.2083$
14	16	$15.2587$	$0.7413$	$15.6336$	$1.041$	$1.0094$
15	13	$12.6634$	$0.3366$	$16.4906$	$1.052$	$0.7823$
16	17	$17.9523$	$- 0.9523$	$16.8888$	$1.0297$	$1.0238$